Graphs on Generalised Baire Spaces

本课题的研究是在公理化集合理论范围内进行的。这个理论通常被看作是所有数学的基础，因为每一个数学概念都可以用无限集的结构来表达。虽然世界的大小是有限的，但我们的计数数字“N”的无限大的理论影响是通过图灵发现的程序和数字计算模型来感受的：虽然计算机是有限的，但关于它们的能力的理论化最好是在无限的背景下完成的。同样，我们用“无限结构”和理论来模拟有限世界。现代集合论的鼻祖康托尔试图解决关于实数线子集的棘手问题，他首先为简单描述的集合建立结果，然后为更复杂的集合建立结果，等等。这就建立了“描述性集合论”的概念。在20世纪10年代和20年代的“古典时期”，俄罗斯（Suslin, Luzin）和法国（Borel, Lebesgue）学派的分析学家集中精力建立他们可以描述为“Borel”或“分析”集的结果。例如，对于平面上的这些集合，即使它们不是由简单曲线包围的区域，也可以发展出“面积”的概念。然而，事情就停留在这个层面上。勒贝格（Lebesgue）在分析集之外定义了一个“投影集”的层次结构，但他对发现它们是否可以在这个意义上“测量”感到绝望。现代集合论发现了为什么古典分析学家会陷入困境：需要一些公理或假设，而不是Zermelo-Fraenkel（在20世纪20年代的“ZF”中发展起来的）标准使用的公理或假设。要么需要在集合的宇宙中假设更强的“无穷公理”（也称为“大基数”），以使这些射影集表现得适当。一个令人惊讶但重要的发展是使用了无限长的两人完美信息游戏。假设这类游戏中有获胜策略发挥了作用。玩家交替移动整数步，游戏长度与n相同，这些在技术上被称为“贝尔空间游戏”。我们的项目是将这些想法重新集中在当前一个新兴的兴趣领域：“广义贝尔空间”：代替可以被解释为实数的十进制展开的N型序列，我们研究更长的序列，即康托尔大不可数基数之一的类型，它比自然数的大小还要大。从这个意义上说，相关的概念游戏也更长，可能会（也可能不会）受到与早期游戏相同的分析。我们还不知道。原始的贝尔空间通常被认为是无理数（剩余的可数的无理数，不包括面积，测度等概念），因此我们可以认为广义的版本是在这个特定的方向上推广了实数线。我们为什么要担心这个呢？研究这种更强的公理的含义要广泛得多：对于一般的数学分析师来说，强公理会影响他们对实数轴的看法，而这一点直到现在才开始得到重视。纯数学的几个领域可以说直接受到集合论公理化的影响。从更广泛的角度来看，理解“无限”和“集合”的本质在哲学上和人类的一般努力中都是有趣的。因此，我们认为这项研究的受益者主要是集合理论家，但更广泛地说，是对这些问题感兴趣的数学逻辑学家和数学哲学家。集合论在国际上非常活跃，在美国、以色列、奥地利、法国、德国都有重要的研究小组。然而，在英国，先进的集合论在某种程度上被低估了，主要集中在布里斯托尔、东英吉利大学和利兹大学。因此，该项目将提高英国在集合论方面的地位和专业知识。

The research of the proposed project is within axiomatic set theory. This theory is usually seen as a foundation for all of mathematics, since every mathematical concept can be expressed structurally in terms of infinite sets. Although the world is of finite size, the theoretical effects of the infinity of our counting numbers, ``N'', is felt through, eg, modelling of computation by programs and numbers as discovered by Turing: although computers are finite, theorizing about their capabilities is best done in an infinite context. In similar ways we model the finite world by using 'infinite structures' and theories.G. Cantor, the originator of modern set theory, tried to solve knotty problems about subsets of the real number line by establishing results first for simply described sets, then building up for more complicated ones, etc. This founded the concept of 'descriptive set theory'. In the `classical period' of the 1910's and 20's the Russian (Suslin, Luzin) and French (Borel, Lebesgue) schools of analysts worked intensively on establishing results up to the level they could describe: 'Borel' or 'analytic' sets. For example, for these sets in the plane an idea of "area'' can be developed even if these are not regions enclosed by a simple curve. However matters were stuck at this level. Lebesgue had defined a hierarchy of "projective sets'' beyond the analytic, but despaired of discovering whether they could be 'measurable' in this sense. Modern set theory has discovered why the classical analysts were stuck: axioms, or postulates, beyond the standardly used ones of Zermelo-Fraenkel (developed in the 1920's "ZF'') were needed. Either stronger "axioms of infinity" (also called "large cardinals'') were needed to be assumed in the universe of sets to get these projective sets to behave properly. One surprising but significant development was the use of infinitely long two person perfect information games. Assuming such games had winning strategies played a role. Players alternated integer moves, and the games had length the same type as N. These are technically known as "games on Baire space''. Our project is to refocus some of these ideas on a current new area of interest that has sprung up: "Generalised Baire spaces'': instead of sequences of type N that can be construed as a decimal expansion of a real number, we look at yet longer sequences the type of one of Cantor's large uncountable cardinal numbers, that is yet greater than the size of the natural numbers. The associated conceptual games are also longer in this sense, and may, or may not, be susceptible to the same kinds of analysis as the earlier ones. We do not yet know. The original Baire space is often identified with the irrational numbers (the countably many rationals left over not counting towards notions of area, measure etc.) We can thus think of the Generalised versions as generalising the real number line in this particular direction.Why should we be concerned about this? The implication of studying such stronger axioms are much wider: for the general mathematical analysts strong axioms affect how they view the real number line, and this is only now starting to be appreciated. Several areas of pure mathematics can be said to be directly affected by set theoretic axiomatics. In the wider perspective an understanding of the nature of 'infinity' and 'set' is of interest both philosophically and for the general human endeavour. We thus think of the beneficiaries of this research as principally set theorists, but more widely,mathematical logicians and philosophers of mathematics who are interested in these questions.Set Theory is very active internationally, with significant research groups in, eg, USA, Israel, Austria, France, Germany. However, in the UK advanced set theory is somewhat underrepresented, and is concentrated in Bristol, UEA and at Leeds. This project will thus enhance the UK's standing and expertise in set theory.